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We study the radial and non-radial oscillations of Cross stars (CrSs), i.e.,stars with a quark matter crust and a hadronic matter core in an inverted ordercompared to conventional hybrid stars. We draw comparisons of their oscillationmodes with those of neutron stars, quark stars, and conventional hybrid stars.We find that the stellar stability analysis from the fundamental mode of radialoscillations, and the $g$, $f$ modes of non-radial oscillations are quitesimilar to those of conventional hybrid stars. However, due to the invertedstellar structure, the first non-radial $p$ mode of CrSs behaves in an invertedway and sits in a higher frequency domain compared to that of conventionalhybrid stars. These results provide a direct way to discriminate CrSs fromother types of compact stars via gravitational-wave probes. Specifically,compact stars emitting $g$-mode gravitational waves within the $0.5$--$1$ kHzrange should be CrSs or conventional hybrid stars rather than neutron stars orpure quark stars, and a further GW detection of the first $p$ mode above 8 kHzor an identification of a decreasing trend of frequencies versus star massesassociated with it will help identify the compact object to be a CrS ratherthan a conventional hybrid star.

Radial and Non-Radial Oscillations of Inverted Hybrid Stars